Semidirect product of a Lie algebra and a module ($\mathfrak{g}⋉V$)

For a Lie algebra representation of $\mathfrak{g}$ on $V$, the semidirect product $\mathfrak{g}⋉V$ is the vector space $\mathfrak{g}\oplus V$ with bracketThe subspace $V$ is an abelian ideal of a Lie algebra.

For a Lie algebra representation of $\mathfrak{g}$ on $V$, let $\mathfrak{g}V$ denote the linear span of $xv$ for $x\in \mathfrak{g}$ and $v\in V$. The derived algebra of the semidirect product of a Lie algebra and a module is the displayed direct sum of vector spaces. Every Lie bracket has first component in $[\mathfrak{g},\mathfrak{g}]$ and second component in $\mathfrak{g}V$. Conversely, brackets of $(x,0)$ with $(y,0)$ span the former component, and brackets of $(x,0)$ with $(0,v)$ span the latter. Therefore this semidirect product of a Lie algebra and a module is a perfect Lie algebra exactly when $\mathfrak{g}$ is a perfect Lie algebra and $\mathfrak{g}V=V$. When $V\ne 0$, its nonzero abelian Lie algebra ideal prevents it from being a semisimple Lie algebra. The sl2 Lie algebra acting on its two-dimensional defining representation supplies a five-dimensional example.

For a finite-dimensional representation $\rho :\mathfrak{g}\to \mathfrak{gl}(V)$,In particular, $V$ lies in the radical of the Killing form.